# 4.10.2. Principal Component Analysis (PCA) — `MDAnalysis.analysis.pca`

Authors:

  John Detlefs

Year:

  2016

Copyright:

  Lesser GNU Public License v2.1+

Added in version 0.16.0.

This module contains the linear dimensions reduction method Principal Component Analysis (PCA). PCA sorts a simulation into 3N directions of descending variance, with N being the number of atoms. These directions are called the principal components. The dimensions to be analyzed are reduced by only looking at a few projections of the first principal components. To learn how to run a Principal Component Analysis, please refer to the PCA Tutorial.

The PCA problem is solved by solving the eigenvalue problem of the covariance matrix, a \(3N \times 3N\) matrix where the element \((i, j)\) is the covariance between coordinates \(i\) and \(j\). The principal components are the eigenvectors of this matrix.

For each eigenvector, its eigenvalue is the variance that the eigenvector explains. Stored in `PCA.results.cumulated_variance`, a ratio for each number of eigenvectors up to index \(i\) is provided to quickly find out how many principal components are needed to explain the amount of variance reflected by those \(i\) eigenvectors. For most data, `PCA.results.cumulated_variance` will be approximately equal to one for some \(n\) that is significantly smaller than the total number of components. These are the components of interest given by Principal Component Analysis.

From here, we can project a trajectory onto these principal components and attempt to retrieve some structure from our high dimensional data.

For a basic introduction to the module, the PCA Tutorial shows how to perform Principal Component Analysis.

## 4.10.2.1. PCA Tutorial

The example uses files provided as part of the MDAnalysis test suite (in the variables `PSF` and `DCD`). This tutorial shows how to use the PCA class.

First load all modules and test data:

    import MDAnalysis as mda
    import MDAnalysis.analysis.pca as pca
    from MDAnalysis.tests.datafiles import PSF, DCD

Given a universe containing trajectory data we can perform Principal Component Analysis by using the class `PCA` and retrieving the principal components.:

    u = mda.Universe(PSF, DCD)
    PSF_pca = pca.PCA(u, select='backbone')
    PSF_pca.run()

Inspect the components to determine the principal components you would like to retain. The choice is arbitrary, but I will stop when 95 percent of the variance is explained by the components. This cumulated variance by the components is conveniently stored in the one-dimensional array attribute `PCA.results.cumulated_variance`. The value at the ith index of `PCA.results.cumulated_variance` is the sum of the variances from 0 to i.:

    n_pcs = np.where(PSF_pca.results.cumulated_variance > 0.95)[0][0]
    atomgroup = u.select_atoms('backbone')
    pca_space = PSF_pca.transform(atomgroup, n_components=n_pcs)

From here, inspection of the `pca_space` and conclusions to be drawn from the data are left to the user.

## 4.10.2.2. Classes and Functions

class MDAnalysis.analysis.pca.PCA(universe, select='all', align=False, mean=None, n_components=None, **kwargs)

  Principal component analysis on an MD trajectory.

  After initializing and calling method with a universe or an atom group, principal components ordering the atom coordinate data by decreasing variance will be available for analysis. As an example::

      pca = PCA(universe, select='backbone').run()
      pca_space = pca.transform(universe.select_atoms('backbone'), 3)

  generates the principal components of the backbone of the atomgroup and then transforms those atomgroup coordinates by the direction of those variances. Please refer to the PCA Tutorial for more detailed instructions. When using mean selections, the first frame of the selected trajectory slice is used as a reference.

  Parameters:

    - universe (Universe) – Universe

    - select (string, optional) – A valid selection statement for choosing a subset of atoms from the atomgroup.

    - align (boolean, optional) – If True, the trajectory will be aligned to a reference structure.

    - mean (array_like, optional) – Optional reference positions to be be used as the mean of the covariance matrix.

    - n_components (int, optional) – The number of principal components to be saved, default saves all principal components

    - verbose (bool (optional)) – Show detailed progress of the calculation if set to `True`.

  results.p_components

    Principal components of the feature space, representing the directions of maximum variance in the data. The column vector p_components[:, i] is the eigenvector corresponding to the variance[i].

    Added in version 2.0.0.

    Type:

      array, (n_atoms * 3, n_components)

  p_components

    Alias to the `results.p_components`.

    Deprecated since version 2.0.0: Will be removed in MDAnalysis 3.0.0. Please use `results.p_components` instead.

    Type:

      array, (n_atoms * 3, n_components)

  results.variance

    Raw variance explained by each eigenvector of the covariance matrix.

    Added in version 2.0.0.

    Type:

      array (n_components, )

  variance

    Alias to the `results.variance`.

    Deprecated since version 2.0.0: Will be removed in MDAnalysis 3.0.0. Please use `results.variance` instead.

    Type:

      array (n_components, )

  results.cumulated_variance

    Percentage of variance explained by the selected components and the sum of the components preceding it. If a subset of components is not chosen then all components are stored and the cumulated variance will converge to 1.

    Added in version 2.0.0.

    Type:

      array, (n_components, )

  cumulated_variance

    Alias to the `results.cumulated_variance`.

    Deprecated since version 2.0.0: Will be removed in MDAnalysis 3.0.0. Please use `results.cumulated_variance` instead.

    Type:

      array, (n_components, )

  Notes

  Computation can be sped up by supplying precalculated mean positions.

  Changed in version 0.19.0: The start frame is used when performing selections and calculating mean positions. Previously the 0th frame was always used.

  Changed in version 1.0.0: `n_components` now limits the correct axis of `p_components`. `cumulated_variance` now accurately represents the contribution of each principal component and does not change when `n_components` is given. If `n_components` is not None or is less than the number of `p_components`, `cumulated_variance` will not sum to 1. `align=True` now correctly aligns the trajectory and computes the correct means and covariance matrix.

  Changed in version 2.0.0: `mean_atoms` removed, as this did not reliably contain the mean positions. `mean` input now accepts coordinate arrays instead of atomgroup. `p_components`, `variance` and `cumulated_variance` are now stored in a `MDAnalysis.analysis.base.Results` instance.

  Changed in version 2.8.0: `self.run()` can now appropriately use `frames` parameter (bug described by #4425 and fixed by #4423). Previously, behaviour was to manually iterate through `self._trajectory`, which would incorrectly handle cases where the `frame` argument was passed.

  cumulative_overlap(other, i=0, n_components=None)

    Compute the cumulative overlap of a vector in a subspace.

    This is not symmetric. The cumulative overlap measures the overlap of the chosen vector in this instance, in the `other` subspace.

    Please cite [Yang2008] if you use this function.

    Parameters:

      - other (`PCA`) – Another PCA class. This must have already been run.

      - i (int, optional) – The index of eigenvector to be analysed.

      - n_components (int, optional) – number of components in `other` to compute for the cumulative overlap. `None` computes all of them.

    Returns:

      Cumulative overlap of the chosen vector in this instance to the `other` subspace. 0 indicates that they are mutually orthogonal, whereas 1 indicates that they are identical.

    Return type:

      float

    See also

    `cumulative_overlap()`

    Added in version 1.0.0.

  classmethod get_supported_backends()

    Tuple with backends supported by the core library for a given class. User can pass either one of these values as `backend=...` to `run()` method, or a custom object that has `apply` method (see documentation for `run()`):

    > - ‘serial’: no parallelization

    > - ‘multiprocessing’: parallelization using multiprocessing.Pool

    > - ‘dask’: parallelization using dask.delayed.compute(). Requires installation of mdanalysis[dask]

    If you want to add your own backend to an existing class, pass a `backends.BackendBase` subclass (see its documentation to learn how to implement it properly), and specify `unsupported_backend=True`.

    Returns:

      names of built-in backends that can be used in `run(backend=...)()`

    Return type:

      tuple

    Added in version 2.8.0.

  property parallelizable

    Boolean mark showing that a given class can be parallelizable with split-apply-combine procedure. Namely, if we can safely distribute `_single_frame()` to multiple workers and then combine them with a proper `_conclude()` call. If set to `False`, no backends except for `serial` are supported.

    Note

    If you want to check parallelizability of the whole class, without explicitly creating an instance of the class, see `_analysis_algorithm_is_parallelizable`. Note that you setting it to other value will break things if the algorithm behind the analysis is not trivially parallelizable.

    Returns:

      if a given `AnalysisBase` subclass instance is parallelizable with split-apply-combine, or not

    Return type:

      bool

    Added in version 2.8.0.

  project_single_frame(components=None, group=None, anchor=None)

    Computes a function to project structures onto selected PCs

    Applies Inverse-PCA transform to the PCA atomgroup. Optionally, calculates one displacement vector per residue to extrapolate the transform to atoms not in the PCA atomgroup.

    Parameters:

      - components (int, array, optional) – Components to be projected onto. The default `None` maps onto all components.

      - group (AtomGroup, optional) – The AtomGroup containing atoms to be projected. The projection applies to whole residues in `group`. The atoms in the PCA class are not affected by this argument. The default `None` does not extrapolate the projection to non-PCA atoms.

      - anchor (string, optional) – The string to select the PCA atom whose displacement vector is applied to non-PCA atoms in a residue. The `anchor` selection is applied to `group`.The resulting atomselection must have exactly one PCA atom in each residue of `group`. The default `None` does not extrapolate the projection to non-PCA atoms.

    Returns:

      The resulting function f(ts) takes as input a `Timestep` ts, and returns ts with the projected structure

      Warning

      The transformation function takes a `Timestep` as input because this is required for Trajectory transformations (“on-the-fly” transformations). However, the inverse-PCA transformation is applied on the atoms of the Universe that was used for the PCA. It is expected that the ts is from the same Universe but this is currently not checked.

    Return type:

      function

    Notes

    When the PCA class is run for an atomgroup, the principal components are cached. The trajectory can then be projected onto one or more of these principal components. Since the principal components are sorted in the order of decreasing explained variance, the first few components capture the essential molecular motion.

    If N is the number of atoms in the PCA group, each component has the length 3N. A PCA score \(w\_i\), along component \(u\_i\), is calculated for a set of coordinates \((r(t))\) of the same atoms. The PCA scores are then used to transform the structure, \((r(t))\) at a timestep, back to the original space.

    \[\begin{split}w_{i}(t) = ({\textbf r}(t) - \bar{{\textbf r}}) \cdot {\textbf u}_i \\ {\textbf r'}(t) = (w_{i}(t) \cdot {\textbf u}_i^T) + \bar{{\textbf r}}\end{split}\]

    For each residue, the projection can be extended to atoms that were not part of PCA by applying the displacement vector of a PCA atom to all the atoms in the residue. This could be useful to preserve the bond distance between a PCA atom and other non-PCA atoms in a residue.

    If there are r residues and n non-PCA atoms in total, the displacement vector has the size 3r. This needs to be broadcasted to a size 3n. An extrapolation trick is used to shape the array, since going over each residue for each frame can be expensive. Non-PCA atoms’ displacement vector is calculated with fancy indexing on the anchors’ displacement vector. index_extrapolate saves which atoms belong to which anchors. If there are two non-PCA atoms in the first anchor’s residue and three in the second anchor’s residue, index_extrapolate is [0, 0, 1, 1, 1]

    Examples

    Run PCA class before using this function. For backbone PCA, run:

        pca = PCA(universe, select='backbone').run()

    Obtain a transformation function to project the backbone trajectory onto the first principal component:

        project = pca.project_single_frame(components=0)

    To project onto the first two components, run:

        project = pca.project_single_frame(components=[0,1])

    Alternatively, the transformation can be applied to PCA atoms and extrapolated to other atoms according to the CA atom’s translation in each residue:

        all = u.select_atoms('all')
        project = pca.project_single_frame(components=0,
                                           group=all, anchor='name CA')

    Finally, apply the transformation function to a timestep:

        project(u.trajectory.ts)

    or apply the projection to the universe:

        u.trajectory.add_transformations(project)

    Added in version 2.2.0.

  rmsip(other, n_components=None)

    Compute the root mean square inner product between subspaces.

    This is only symmetric if the number of components is the same for both instances. The RMSIP effectively measures how correlated the vectors of this instance are to those of `other`.

    Please cite [Amadei1999] and [Leo-Macias2004] if you use this function.

    Parameters:

      - other (`PCA`) – Another PCA class. This must have already been run.

      - n_components (int or tuple of ints, optional) – number of components to compute for the inner products. `None` computes all of them.

    Returns:

      Root mean square inner product of the selected subspaces. 0 indicates that they are mutually orthogonal, whereas 1 indicates that they are identical.

    Return type:

      float

    Examples

    You can compare the RMSIP between different intervals of the same trajectory. For example, to compare similarity within the top three principal components:

        >>> first_interval = pca.PCA(u, select="backbone").run(start=0, stop=25)
        >>> second_interval = pca.PCA(u, select="backbone").run(start=25, stop=50)
        >>> last_interval = pca.PCA(u, select="backbone").run(start=75)
        >>> round(first_interval.rmsip(second_interval, n_components=3), 6)
        0.381476
        >>> round(first_interval.rmsip(last_interval, n_components=3), 6)
        0.174782

    See also

    `rmsip()`

    Added in version 1.0.0.

  run(start: int = None, stop: int = None, step: int = None, frames: Iterable = None, verbose: bool = None, n_workers: int = None, n_parts: int = None, backend: str | BackendBase = None, *, unsupported_backend: bool = False, progressbar_kwargs=None)

    Perform the calculation

    Parameters:

      - start (int, optional) – start frame of analysis

      - stop (int, optional) – stop frame of analysis

      - step (int, optional) – number of frames to skip between each analysed frame

      - frames (array_like, optional) –

        array of integers or booleans to slice trajectory; `frames` can only be used instead of `start`, `stop`, and `step`. Setting both `frames` and at least one of `start`, `stop`, `step` to a non-default value will raise a `ValueError`.

        Added in version 2.2.0.

      - verbose (bool, optional) – Turn on verbosity

      - progressbar_kwargs (dict, optional) – ProgressBar keywords with custom parameters regarding progress bar position, etc; see `MDAnalysis.lib.log.ProgressBar` for full list. Available only for `backend='serial'`

      - backend (Union[str, BackendBase], optional) –

        By default, performs calculations in a serial fashion. Otherwise, user can choose a backend: `str` is matched to a builtin backend (one of `serial`, `multiprocessing` and `dask`), or a `MDAnalysis.analysis.results.BackendBase` subclass.

        Added in version 2.8.0.

      - n_workers (int) –

        positive integer with number of workers (processes, in case of built-in backends) to split the work between

        Added in version 2.8.0.

      - n_parts (int, optional) –

        number of parts to split computations across. Can be more than number of workers.

        Added in version 2.8.0.

      - unsupported_backend (bool, optional) –

        if you want to run your custom backend on a parallelizable class that has not been tested by developers, by default False

        Added in version 2.8.0.

    Changed in version 2.2.0: Added ability to analyze arbitrary frames by passing a list of frame indices in the frames keyword argument.

    Changed in version 2.5.0: Add progressbar_kwargs parameter, allowing to modify description, position etc of tqdm progressbars

    Changed in version 2.8.0: Introduced `backend`, `n_workers`, `n_parts` and `unsupported_backend` keywords, and refactored the method logic to support parallelizable execution.

  transform(atomgroup, n_components=None, start=None, stop=None, step=None, verbose=False)

    Apply the dimensionality reduction on a trajectory

    Parameters:

      - atomgroup (AtomGroup or Universe) – The AtomGroup or Universe containing atoms to be PCA transformed.

      - n_components (int, optional) – The number of components to be projected onto. The default `None` maps onto all components.

      - start (int, optional) – The frame to start on for the PCA transform. The default `None` becomes 0, the first frame index.

      - stop (int, optional) – Frame index to stop PCA transform. The default `None` becomes the total number of frames in the trajectory. Iteration stops before this frame number, which means that the trajectory would be read until the end.

      - step (int, optional) – Include every step frames in the PCA transform. If set to `None` (the default) then every frame is analyzed (i.e., same as `step=1`).

      - verbose (bool, optional) – `verbose = True` option displays a progress bar for the iterations of transform. `verbose = False` disables the progress bar, just returns the pca_space array when the calculations are finished.

    Returns:

      pca_space

    Return type:

      array, shape (n_frames, n_components)

    Changed in version 0.19.0: Transform now requires that `run()` has been called before, otherwise a `ValueError` is raised.

    Changed in version 2.8.0: Transform now has shows a tqdm progressbar, which can be toggled on with `verbose = True`, or off with `verbose = False`

MDAnalysis.analysis.pca.cosine_content(pca_space, i)

  Measure the cosine content of the PCA projection.

  The cosine content of pca projections can be used as an indicator if a simulation is converged. Values close to 1 are an indicator that the simulation isn’t converged. For values below 0.7 no statement can be made. If you use this function please cite [1].

  Parameters:

    - pca_space (array, shape (number of frames, number of components)) – The PCA space to be analyzed.

    - i (int) – The index of the pca_component projection to be analyzed.

  Returns:

    - A float reflecting the cosine content of the ith projection in the PCA

    - space. The output is bounded by 0 and 1, with 1 reflecting an agreement

    - with cosine while 0 reflects complete disagreement.

  References

  [1]

  Berk Hess. Convergence of sampling in protein simulations. Phys. Rev. E, 65:031910, Mar 2002. doi:10.1103/PhysRevE.65.031910.

MDAnalysis.analysis.pca.rmsip(a, b, n_components=None)

  Compute the root mean square inner product between subspaces.

  This is only symmetric if the number of components is the same for `a` and `b`. The RMSIP effectively measures how correlated the vectors of `a` are to those of `b`.

  Please cite [Amadei1999] and [Leo-Macias2004] if you use this function.

  Parameters:

    - a (array, shape (n_components, n_features)) – The first subspace. Must have the same number of features as `b`. If you are using the results of `PCA`, this is the TRANSPOSE of `p_components` (i.e. `p_components.T`).

    - b (array, shape (n_components, n_features)) – The second subspace. Must have the same number of features as `a`. If you are using the results of `PCA`, this is the TRANSPOSE of `p_components` (i.e. `p_components.T`).

    - n_components (int or tuple of ints, optional) – number of components to compute for the inner products. `None` computes all of them.

  Returns:

    Root mean square inner product of the selected subspaces. 0 indicates that they are mutually orthogonal, whereas 1 indicates that they are identical.

  Return type:

    float

  Examples

  You can compare the RMSIP between different intervals of the same trajectory. For example, to compare similarity within the top three principal components:

      >>> first_interval = pca.PCA(u, select="backbone").run(start=0, stop=25)
      >>> second_interval = pca.PCA(u, select="backbone").run(start=25, stop=50)
      >>> last_interval = pca.PCA(u, select="backbone").run(start=75)
      >>> round(pca.rmsip(first_interval.results.p_components.T,
      ...           second_interval.results.p_components.T,
      ...           n_components=3), 6)
      0.381476
      >>> round(pca.rmsip(first_interval.results.p_components.T,
      ...           last_interval.results.p_components.T,
      ...           n_components=3), 6)
      0.174782

  Added in version 1.0.0.

MDAnalysis.analysis.pca.cumulative_overlap(a, b, i=0, n_components=None)

  Compute the cumulative overlap of a vector in a subspace.

  This is not symmetric. The cumulative overlap measures the overlap of the chosen vector in `a`, in the `b` subspace.

  Please cite [Yang2008] if you use this function.

  Parameters:

    - a (array, shape (n_components, n_features) or vector, length n_features) – The first subspace containing the vector of interest. Alternatively, the actual vector. Must have the same number of features as `b`.

    - b (array, shape (n_components, n_features)) – The second subspace. Must have the same number of features as `a`.

    - i (int, optional) – The index of eigenvector to be analysed.

    - n_components (int, optional) – number of components in `b` to compute for the cumulative overlap. `None` computes all of them.

  Returns:

    Cumulative overlap of the chosen vector in `a` to the `b` subspace. 0 indicates that they are mutually orthogonal, whereas 1 indicates that they are identical.

  Return type:

    float

  Added in version 1.0.0.